[ben_w]

That example is completely false: how much rain will fall is absolutely a computable function, just a very difficult and expensive function to evaluate with absurdly large boundary conditions.

This is in the same sense that while it is technically correct to describe all physically instantiated computer programs, and by extension all AI, as being in the set of "things which are just Markov chains", it comes with a massive cost that may or may not be physically realisable within this universe.

Rainfall to the exact number of molecules is computable. Just hard. A quantum simulation of every protein folding and every electron energy level of every atom inside every cell of your brain on a classical computer is computable, in the Church-Turing sense, just with an exponential slowdown.

The busy beaver function, however, is actually un-computable.

[Verdex]

The busy beaver function isn't uncomputable.

You just compute the brains of a bunch of immortal mathematics. At which point it's "very difficult and expensive function to evaluate with absurdly large boundary conditions."

[ben_w]

> The busy beaver function isn't uncomputable.

False.

To quote:

  One of the most consequential aspects of the busy beaver game is that, if it were possible to compute the functions Σ(n) and S(n) for all n, then this would resolve all mathematical conjectures which can be encoded in the form "does ⟨this Turing machine⟩ halt".[5] For example, there is a 27-state Turing machine that checks Goldbach's conjecture for each number and halts on a counterexample; if this machine did not halt after running for S(27) steps, then it must run forever, resolving the conjecture.[5][7] Many other problems, including the Riemann hypothesis (744 states) and the consistency of ZF set theory (745 states[8][9]), can be expressed in a similar form, where at most a countably infinite number of cases need to be checked.[5]

"Uncomputable" has a very specific meaning, and the busy beaver function is one of those things, it is not merely "hard".

> You just compute the brains of a bunch of immortal mathematics. At which point it's "very difficult and expensive function to evaluate with absurdly large boundary conditions."

Humans are not magic, humans cannot solve it either, just as they cannot magically solve the halting problem for all inputs.